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How the result of graph clustering methods depends on the construction of the graph

Published online by Cambridge University Press:  21 May 2013

Markus Maier ,
Ulrike von Luxburg  and
Matthias Hein
Markus Maier
Affiliation:
Max Planck Institute for Intelligent Systems, Spemannstr. 38, 72076 Tübingen, Germany. mmaier@tuebingen.mpg.de
Ulrike von Luxburg
Affiliation:
Max Planck Institute for Intelligent Systems, Spemannstr. 38, 72076 Tübingen, Germany. mmaier@tuebingen.mpg.de Department of Computer Science, University of Hamburg, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany; luxburg@informatik.uni-hamburg.de
Matthias Hein
Affiliation:
Faculty of Mathematics and Computer Science, Saarland University, Postfach 151150, 66041, Saarbrücken, Germany; hein@cs.uni-sb.de
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Abstract

We study the scenario of graph-based clustering algorithms such as spectral clustering. Given a set of data points, one first has to construct a graph on the data points and then apply a graph clustering algorithm to find a suitable partition of the graph. Our main question is if and how the construction of the graph (choice of the graph, choice of parameters, choice of weights) influences the outcome of the final clustering result. To this end we study the convergence of cluster quality measures such as the normalized cut or the Cheeger cut on various kinds of random geometric graphs as the sample size tends to infinity. It turns out that the limit values of the same objective function are systematically different on different types of graphs. This implies that clustering results systematically depend on the graph and can be very different for different types of graph. We provide examples to illustrate the implications on spectral clustering.

Keywords

Random geometric graphclusteringgraph cuts
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 370 - 418
Copyright
© EDP Sciences, SMAI, 2013

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